Roof construction

ABSTRACT

A roof structure having a peripheral support ring is disclosed wherein the ring conforms in plan substantially to a closed curve having major and minor axes and at least two skewed axes of symmetry. A plurality of sets of arches are connected to the ring to form the roof, with the arches of at least two sets respectively extending in plan substantially parallel to a separate one of the skewed axes of symmetry of the closed curve and the arches of another set extending in plan substantially parallel to the major and/or the minor axes of the closed curve. The arches impose a funicular load on the ring and support a roof deck structure to form a dome surface.

United States Geiger [45] Nm. 241], i973 [54] ROOF CONSTRUCTION [76]Inventor: David H. Geiger, 788 Riverside Dr.,

New York, N Y. 10032 [22] Filed; Api-.12,1971

[21] Appl. No.: 133,198

FOREIGN PATENTS OR APPLICATIONS 197,921 8/1967 U.S.S.R 52/83 664,566 6/1964 Italy 52/80 1,191,776 10/1959 France.... 52/81 484,436 5/1925Germany 52/86 415,870 7/1925 Germany 52/86 61,853 5/1968 Germany 52/86OTHER PUBLICATIONS Curved Roof on Cables Spans Big Arena; EngineeringNews Record Feb. 5, 1953 pages 31-37.

British Build Largest Dome, Engineering News Record Oct. 5, 1950 page33.

Elliptical Domes, Dome Book Il May, 1971 pages 35-39.

Primary Examiner-Frank L. Abbott Assistant Examiner-H. E. RaduazoAttorney-Curtis, Morris & Safford [57] ABSTRACT A roof structure havinga peripheral support ring is disclosed wherein the ring conforms in plansubstantially to a closed curve having major and minor axes and at leasttwo skewed axes of symmetry. A plurality of sets of arches are connectedto the ring to form the roof, with the arches of at least two setsrespectively extending in plan substantially parallel to a separate oneof the skewed axes of symmetry of the closed curve and the arches ofanother set extending in plan substantially parallel to the major and/orthe minor axes of the closed curve. The arches impose a funicular loadon the ring and support a roof deck structure to form a dome surface.

8 Claims, 8 Drawing Figures PAIENEuuv 2o ma 3,772,836 SHEET 1 nr 5INVENTOR ATTOR NEYS RGOF CONSTRUCTIUN This application is acontinuation-in-part of copending U. S. Pat. application Ser. No.80,048, tiled Oct. l2, 1970 the disclosure of which is incorporatedherein by reference.

This invention relates to a roof construction, and

more particularly to a domed roof structure including a peripheralstructural ring which restrains the remaining elements of the roof.

In dome roof constructions wherein the roof is supported only by thestructural members forming the dome, without the use of interior columnsor beams, it is desirable to control the nature of forces in thestructural elements of the dome so that each of the structural elementscarry nearly the same load and may be formed of the same size structuralmember, or so that joint details, particularly where the joint is formedof wood, may be readily solved for transmission of bearing forcestherethrough.

The dome of the present invention accomplishes these and other ends byutilizing a peripheral ring and structural elements such as rigidarches, which are connected to the ring in a predetermined pattern suchthat the ring is loaded in a substantially funicular manner. Theconfiguration of the ring is selected from the family of closed curveshaving major and minor axes and at least two skewed axes of symmetry(more fully described hereinafter) and a plurality of sets of arches aresecured to the ring for supporting the roof deck structure. Two of thesets of arches are respectively positioned parallel to the skewed axesof symmetry of the ring and a third set of arches are located in plansubstantially parallel to the major or minor axes of the ring and passthrough the vertices of the diamond shaped pattern generated by thefirst two sets of arches. In this manner the framing system for therigid structure is triangulated, except possibly at the ring.Triangulation at the ring is accomplished, if desired, by slight changesin directions of one or more of the arches adjacent the ring, to enhancestructural stability of the dome.

The above, and other features and advantages of this invention, will beapparent in the following detailed description of illustrativeembodiments thereof which are to be read in connection with theaccompanying drawings, wherein:

FIG. l is a diagram of a closed curve to which the ring of the roofstructure of the present invention may conform and illustrates certainproperties of that curve useful in explaining the invention;

FIG. 2 is a diagram showing the family of superellipses (x/a)'" (y/b) l,with a a b but of the same values for each ellipse shown and with mpassing through values (including non-integral values) from less thanunity up to infinity, for which latter value the ellipse takes the formof a rectangle circumscribing all ellipses of the family;

FIG. 3 is a schematic plan view of a roof construction according to theinvention;

FIG..4 is a force diagram of a portion of the ring structure shown inFIG. 3;

FIGS. 5 and 6 are schematic plan views of other roof structuresconstructed according to the present invention;

FIG. '7 is a sectional view taken along line 7-'7 of FIG. 6 illustratingthe foundation support of the ring for a roof structure constructed inaccordance with the present invention; and

FIG. is a sectional view taken along line 8 8 of FIG. 7.

Referring to the drawings in detail, and initially to FIG. l thereof,there are illustrated two line M and N which are not perpendicular andwhich are axes of skewed symmetry for the closed curve l0, since anyline L intersecting one of these axes (for example, line L intersectingaxis N at B) which is parallel to the other axis (M) intersects thecurve l0 at points C and A such that the distance AB is equal to thedistance BC. Curves illustrating this characteristic are referred toherein as curves having skewed axes of symmetry. There are numerouscurves having this characteristic and the curve l0 illustrated in FIG.ll, which is shown by way of example is an ellipse conforming to theusual equation (x/a)2 (y/b)2 l A family of closed curves which hasskewed axes of symmetry, thereby having the characteristic of skewedsymmetry as referred to hereinafter, is the family of ellipses andsuperellipses generated by the equation (x/a)'" -l- (y/b)" l, some ofwhich are shown in FIG. 2. Whatever the value of rn, the ellipse can becircumscribed with a rectangle whose sides are perpendicular to themajor and minor axes of the ellipse, with the rectangle being tangent tothe ellipse at the intersections of the major and minor axes thereof. Itis a property of this family of curves that the diagonals 114 and 16 ofthe circumscribing rectangle )l2 are their axes of skewed symmetry,whatever the value of m. This may be demonstrated by a coordinatetransformation to the g and fr; axis where x= (1; cos a, y (1; f) cos a,and a tanl b/a.

In these expressions, illustrated with reference to FIG. 2, a is theangle between the major axes of the ellipses of that figure and theadjacent diagonal ofthe circumscribing rectangle 12. 'n is the lengthmeasured along the direction of diagonal liti as a slant coordinateaxes, and is the length measured along the diagonal I4 of thecircumscribing rectangle as another slant cordinate axis.

In accordance with the present invention, structural members forming thedome or roof construction are placed so as to project inplan as straightlines parallel to the axes of skewed symmetry of the closed curve, suchas for example, the axes of IVI and N of the ellipse of FIG. ll or theaxes I4 and I6 of the family of ellipses and super ellipses illustratedin FIG. 2. The structual members of the present invention are rigidarches which provide the roof framing structure. With the structuralmembers of the dome positioned in this manner and connected to aperipheral ring conforming, or substantially conforming to a closedcurve having the pro'perty of skewed symmetry, the invention provides aroof structure wherein the ring is substantially free of bending momentsin the horizontal plane and is to that extent funicular. This isachieved by selecting the ordinates of the respective arches such thatthe horizontal component of forces therein, as a result of the roofloads which they must support, impose at their connection with the ringforces which load the ring in a substantially funicular manner.

One roof construction according to the invention is shown in FIG. 3,wherein a ring Il@ of elliptical shape rests upon a suitable foundation20 of concrete or the like, having the same outline as the ring. Twosets of interconnected arches 22 and 24 are provided with the arches ofeach set respectively extending parallel, in projection, onto thehorizontal plane, to one of the diagonals 14 or 16 of the circumscribingrectangle 12. These arches carry the uniform roof load to the ring andapply substantially equal horizontal components of force to the rings attheir opposed ends. Another set of arches 26 are provided in the domeconstruction of the present invention which may extend parallel to themajor or minor axes 30 and 28 respectively, and which are adapted tocarry the unsymmetric and/or antisymmetric loads for which the roof`structure is designed. ln the preferred embodiment of the presentinvention, as illustrated in FIG. 3, the third set of arches 26 arepositioned to extend parallel to the minor axes of the closed curve l0.The arches support a roof decking system (not shown) including purlinsextending between the arches for supporting the roof covering member,which may be of conventional construction, and which forms the domedsurface of the roof structure.

The ordinate location and arrangement of the arches is selectedmathematically so as to control the magnitude and character of theforces within the arches and ring and to provide the roof shape desired.For example, if, as in FIG. 3, the arches are in two sets 22 and 24 ofequally spaced arches parallel to the diagonal axes 14 and 16, thehorizontal components of force to be exerted by each arch at its pointof connection with the ring 18 can be evaluated in terms of the stressalong the ring, i.e., tangent to the axis of the ring, at theintersections of the diagonals 14 and 16 with the ring. The ring willobviously be in tension in the completed roof structure and the tensilestress denoted P, initially selected for the ring on the assumption thatthe arches 22 and 24 are in compression. With P acting at E and F beingthe internal force in the ring, let the horizontal component of the archreactions Hl acting on the ring segment EF (which extends from theintersection of diagonal 16 with the ring at E past the intersectiontherewith of the major axis 30 to the intersection therewith at F of theother diagonal 14) be as shown in FIG. 4. By considering the equilibriumof ring segments of the free body EF from E to H2, E to H3, etc., thehorizontal reactions H1, H2, etc. can be determined in terms of P.Similarly, if there is considered the ring segment DE between theintersection of the ring at D of the opposite end of diagonal 14 and thepoint E already defined, and if there is determined the relationshipbetween the horizontal component Hl and the force P, one finds that theopposite ends of each arch in the set 24 acting on ring segment EF exertthe same horizontal reaction on the ring segment DE. The remainingcorresponding segments of the ring may be similarly analyzed and it isfound that the opposite ends of each of the arches in both sets 22 and24, exert the same horizontal reaction on the ring. Thus, the horizontalcomponent of the forces in the arches required to funicularly load ring18 are determined in terms of the selected force P.

With the horizontal components of the forces in the arches at theirpoints of connection to ring l8 thus determined as a function of theaxial force P in the ring at its intersection with the diagonals 14 and16, a suitable shape for the roof in terms of the rise of the roof abovethe plane of the ring and the consequent configuration of the archesthemselves can be arrived at by considering vertical equilibrium at eachof the intersections of the arches. The material of the roof andaccessory elements such as the arches themselves and the symmetric liveload being known, the weight per unit projected area of the roof domecan be estimated. This gives the vertical force to be supported at thearch intersections (roof joints), and this vertical force with anassumed value of P, can be utilized to compute the vertical ordinates ofthe arches at the various joints necessary for the stresses in the archto have this vertical resultant and to have simultaneously the desiredhorizontal component at the points of connection of the arches with thering. lf the resultant three-dimensional shape of the roof is not theone desired, another value of the axial ring stress P can be assumed andanother shape can be computed for the roof` by the same process. In thisway there can be found at least two roof shapes on either side of theone desired, for example, as to the degree of convexity thereof and thesize of the structural arch members, and extrapolation between the twowill yield the desired shape and the required ring force.

The set 26 of arch members is provided in the dome structure in order toadd structural stability to the dome under unsymmetric or anitsymmetricloads. These arches are positioned in the preferred embodiment of thepresent invention parallel to the minor axes of the ellipse defined byring 18 and are connected, as illustrated in FIG. 3, to the archesextending parallel to the skewed axes of symmetry at the vertices of thediamond-shapped pattern generated by the former set of arches. In thismanner, the framing system of the roof structure is triangulated, exceptpossibly at the ring itself. However, as more fully describedhereinafter, triangulation at the ring may, if desired be accomplishedby slight changes of one or more of the arch directions in theneighborhood of the ring.

The arches in sets 22 and 24 are selected and designed to carry thesymmetrical loading of the roof. The arches in set 26 with the arches insets 22 and 24 carry the antisymmetric loadings for which the roof isdesigned. The relationship between the ring shape, the symmetric roofload and the vertical ordinates of the intersection points of theframing system is established mathematically so as to control themagnitude and size of the forces within the various arch members so thatjoint details in wooden arch structures may be simplified and/or so thatthe arches in steel arch dome structures carry nearly the same load andmay be formed of the same size structural members for economy inconstructing the roof.

For example, the roof ordinates may be established so that the two setsof arches 22 and 24 extending parallel to the axes of skewed symmetryare in compression under the symmetric roof loads and for certain ringshapes and loading these compressive forces may be nearly equal and ofcomparatively large magnitude. Under the symmetric loading case, thethird set of arches 26 will have nearly zero loads. For antisymmetricloads, which are combined with the symmetric loading to determine thestructural design loads (for example full snow load on the other half ofthe structure) it is found that the two sets of arches originally incompression are still in compression. Thus, for the wooden archedstructures, these arches may be made discontinuous at a joint since thecompression forces can be transmitted through joint structures in woodenarch construction in bearing and shear. This permits the other set ofarches 26, which are designed to carry the antisymmetric loads, andwhich may be in tension,

compression, or bending, to be continuous through the joint, since inwood joint constructions it is difficult to transmit tensile or bendingstresses. Thus, by controlling the nature of forces in the system, ajoint detail, particularly, in wood can be readily solved by one skilledin the art. On the other hand, in steel arch constuctions, the jointsbetween the arches readily transmit either tensile or compressiveforces. Accordingly, the ring shape and roof ordinates may beestablished such that each of the arch members are under substantiallythe same stress and carry nearly the same load so that the arches may beformed of the same sized structural members. This results in asubstantial economy in constructing the arched dome since the variety ofsizes of structural members required for the construction is minimized.

Another closed curved having the property of skewed symmetry, whereinthe horizontal forces at the ends of each of the arches extendingparallel to the skewed axis of symmetry have the same value so that theperipheral ring to which the arches are connected is micularly loaded,is illustrated in FIG. 4. The arch structure schematically shown thereinis for a wooden arch construction and the diagonals 32 and 34 of thecircumscribing rectangle 36 are its axes of skewed symmetry while axes38 and 40 are its axes of symmetry in the conventional sense andconstitute minor and major axes respectively. y

The curve illustrated in FIG. S is defined by the set of equations 4.f/d=b/a with a and b having given values equal to one-half the lengthsof axes 40 and 38 respectively, and the values c,d, and n arbitrarilyselected. The ring 50, having dimensions selected in accordance with theabove equations, is arranged to pass through the following coordinateswith respect to the major axis 49 (x) and the minor axis 38 (y):

Each of the coordinates defined in this manner locates on ring 50 ajoint between the ring and the end of one of the arches extendingparallel to the axes of skewed symmetry. By arranging the archesparallel to the skewed axes of symmetry in this manner and with thiscurve, the ring 50 is funicularly loaded so that it is free of bendingmoments in its horizontal plane.

It is noted that a curve constructed in accordance with the aboveequations is also suitable for use with an inflated dome structuresimilar to that disclosed in my above-mentioned copending U.S. patentapplication and that the cables in such a dome structure willfunicularly load the ring to which they are secured. However, the domestructure illustrated in FIG. 5 has been designed for a wooden archconstruction and accordingly, a set of cross-arches 52 extendingparallel to minor axes 38 have been provided.

It is noted that the points of intersection of the arches in sets 56 and58, that is, the vertices of the diamond shaped pattern generated bythese intersecting arches are located along imaginary lines extendingparallel to the major and minor axes of ring 50. These arches areoperatively interconnected at each of these intersections and the latterare located at vertical ordinates with respect to the base, in themanner described above, such that the loading of ring S0 is funicular.Arches 52, provided parallel to minor axis 33, are located alongalternate ones of the imaginary joint lines to provide a coarsetriangulation within the dome for added stability. Thus, for example,triangulation is provided between joints 60, 62, and 64 by the archsections 66, 68 and 70 between these joints. Similarly, other joints inthe frame are triangularly related to provide triangulation in the roofstructure.

As with the previously discussed embodiment, the coarse arches 52 carrythe antisymmetrical loads for which the roof structure is designed.

Each of the joint structures in this dome are illustrated by symbolswhich represent continuity or discontinuity of the respective archesthrough the joint. Thus, since the arches 56 and 58 extend parallel tothe skewed axis of symmetry and are in compression under substantiallyall loading conditions, they are formed in segments between the jointsbecause the compressive forces therein are readily transmitted throughwood joint constructions. On the other hand, the arches in set S2, whichcarry compressive and tensile loads since they are designed to carry theantisymmetric design loads of the roof structure, are continuous atthose joints through which tensile forces must be transmitted. This isadvantageous since, as mentioned above, it is difficult to design woodjoints for transmitting tensile stresses. Accordingly, the problem isavoided by making the arch continuous through such joints.

It is noted that while a coarsetriangulation has been used in theembodiment of the invention illustrated in FIG. 5, a nger triangulation,that is, a dome structure having cross-arches from set 52 through eachof the imaginary lines of joints extending parallel to minor axis 38,may be provided for additional stability against extreme conditions orfor larger and heavier roofs. One such embodiment is illustrated in FIG.6, wherein a dome structure is shown for construction with steelmembers. The dome therein is constructed in accordance with the sameequations and coordinate locations as the previously discussed domewith, however, the arches 52 being located along each of the lines ofthe joints. Further, in this embodiment the triangulation of the archeshas been carried up to ring 50 by deflecting the end portions of certainof the crossarches in set 52. For example, the ends of arches 72, 74, 76and 7% have been slightly diverted from positions parallel to minor axis36 in order to join with an adjacent arch extending parallel to a skewedaxis of symmetry at the ring, so that triangulation is carried to thering.

In the steel construction such as that illustrated in FIG. 6, whereincompressive and tensile forces may be readily transmitted through thejoints between the arches, the ring shape and vertical ordinate of thelocation of the joints between the arches may be determined, asdescribed above, such that the forces along an individual arch will besubstantially equal and so that the forces within sets of arches willalso be substantially equal, whereby all of the arches in a set may beformed of substantially the same sized structural member. In one steeldome structure designed in accordance with the present invention,substantially all of the arches extending parallel to the skewed axes ofsymmetry, that is, the arches in sets 56 and 58, were formed of 10WF 49structural steel members, while the arches in set 52 were formed ofsubstantially all 14 B 26 steel structural members. Thus, a substantialeconomic savings is achieved because there is very little variation inthe type of structural members required for the roof construction.

It is noted, that while each of the above described roof structures,whether formed of rigid arch constructions or of cable supportmembranes, have utilized a perimeter ring which lies in a horizontalplane, the ring may in fact have a variety of configurations ane neednot be horizontal throughout its entire extent. The only limitation onthe configuration of the ring in order to maintain the funicular loadingthereof with structural members positioned as discussed above, is thatthe ring project in plan to a closed curve having skewed axes ofsymmetry, and preferably that the ends of each individual structuralmember be on the same horizontal plane. Thus, for example, the ring maytake the shape, in a side view, of a curve or catenary or any other typeof non-linear configuration so long as the ends of each individual archare on the same level.

Referring now to FIGS. 7 and 8 of the drawing, there is illustrated atypical joint construction and foundation support for the ring 50 of asteel dome construction formed in accordance with the configurationillustrated in FIG. 6. As seen therein, ring 50 is formed of an I orwide flange beam 90 having vertically extending plates 92 securedthereto, at the point of connection between the ring and the arches. Theinside plate 92 has a plate 94 welded thereto and the latter isconnected through shear plate 96 by bolts 98 to the arch at the joint.The arch carries the purlins and decking system schematicallyillustrated at 100 which are secured to the arches in a conventionalmanner. A peripheral gutter system 102 is also secured to the beam 90 toreceive rajn, snow ane the like from the roof and carry it therefrom.

The base of the beams 90 forming ring 50, at each of the joints betweenthe ring and the arch, are welded to a support plate 104 which in turnis welded to a low friction bearing plate 106 having a bearing surface108. Such a plate may, for example, be formed of a fluorogold slidebearing plate or the equivalent. Plate 106 rests on another bearingplate 110 formed of the same material and having a bearing surface 112.Plate 110 is secured, as by welding, to a support plate l 13 anchored ina concrete foundation 114 which extends entirely about the periphery ofthe roof structure. In this manner the ring 50 and roof structure issupported on a concrete foundation and relative movement between thering and the foundation is permitted in order to avoid the transmissionof excessive stresses to the foundation. The bearing plates permitaccommodation by the foundation structure of the expansion andcontraction of the roof structure in accordance with temperaturechanges.

Although illustrative embodiments of the present invention have beendescribed herein with reference to the accompanying drawings, it is tobe understood that the invention is not limited to those preciseembodiments, and that various changes and modifications may be effectedtherein by one skilled in the art without departing from the scope orspirit of this invention.

What is claimed is:

l. A roof structure comprisng a ring projecting in plan substantially toa closed curve having major and minor axes and at least two skewed axesof symmetry, a plurality of sets of relatively rigid arches connected tosaid ring with the arches at at least two sets respectively extending inplan substantially parallel to a separate one of said skewed axes ofsymmetry, and another set of arches extending in plan substantiallyparallel to at least one of said major and minor axes, whereby said ringis funicularly loaded under substantially all loading conditions, saidclosed curve being selected from a family of closed curves defined bythe equations:

4 f/d=b/a which pass through the following coordinates with respect tothe major (x) and minor (y) thereof:

1d+ rr+ nn +2+1)] 'y=ff f,

with a and b being given and equal respectively to onehalf of thepresented length of the major and minor axes of the curve, and c, d, andn being arbitrarily selected.

2. A roof structure as defined in claim 1 wherein each of saidcoordinates defines the location on said ring of the joint between saidring and at least one of said arches extending parallel to said skewedaxes of symmetry.

3. A roof structure projecting in plan substantially to a closed curvehaving major and minor axes and at least two skewed axes of symmetry, aplurality of sets of structural members connected to said ring with themembers of each set respectively extending in plan substantiallyparallel to a separate one of said skewed axes of symmetry, said curvebeing selected from a family of closed curves defined by the equations:

4. f/d=b/a which pass through the following coordinates with respect tothe major (x) and minor (y) axes thereof:

7'. A roof structure as defined in claim 4i wherein said structuralmembers are rigid structural arches.

8. A roof structure as defined in claim 7 including another set ofarches extending substantially parallel to at least one of the major andminor axes of said curve.

1k ik 2l# UNITED STATES PATENT -OFFICE C ERT l |91 C AT E O F C O `lil RE w.'l` Il() N Patent NO- 3.772.836 Dated 1\Iov=.-mb=:r 20. 1973Inventor(s) David H. Geiger It is certified that error appears in theabove-identified patent and that said Letters Patent are herebycorrected as shown below:

Claim` l2 line 6 change "at" (first occurence) tof-of".

Signed and sealed this lith day of May 197k.

(SEAL) Attest:

' EDWARD PLFLETCHEILJR. i C. JIARSHALL DANN Attesting OfficerCommissioner of Patents FORM P04050 (10'59) I uscoMM-Dc eov'e-Pan U. s.GOVERNMENT PRINTING OFFICE t l". 0-36-5,

1. A roof structure comprisng a ring projecting in plan substantially toa closed curve having major and minor axes and at least two skewed axesof symmetry, a plurality of sets of relatively rigid arches connected tosaid ring with the arches at at least two sets respectively extending inplan substantially parallel to a separate one of said skewed axes ofsymmetry, and another set of arches extending in plan substantiallyparallel to at least one of said major and minor axes, whereby said ringis funicularly loaded under substantially all loading conditions, saidclosed curve being selected from a family of closed curves defined bythe equations: 1 a c(1+2+3 . . . (n-1)+n)+d 2 b e(1+2+3 . . . (n-1)+n)+f 3 e/c b/a 4 f/d b/a which pass through the following coordinates withrespect to the major (x) and minor (y) thereof: x + OR - d y + OR -e(1+2+3 . . . (n-1)+n)+f)x + OR (d+nc)y + OR - (e(2+3 . . .(n-1)+n)+f)x + OR - (d+c( n+)n-1))y + OR - (e(3 . . . (n-1)+n)+f). . . .. . x- + OR (d+c(n+(n-1) . . . +2+1))y + OR - f, with a and b beinggiven and equal respectively to one-half of the presented length of themajor and minor axes of the curve, and c, d, and n being arbitrarilyselected.
 2. A roof structure as defined in claim 1 wherein each of saidcoordinates defines the location on said ring of the joint between saidring and at least one of said arches extending parallel to said skewedaxes of symmetry.
 2. b e(1+2+3 . . . (n-1)+n)+f
 3. A roof structureprojecting in plan substantially to a closed curve having major andminor axes and at least two skewed axes of symmetry, a plurality of setsof structural members connected to said ring with the members of eachset respectively extending in plan substantially parallel to a separateone of said skewed axes of symmetry, said curve being selected from afamily of closed curves defined by the equations:
 3. e/c b/a
 4. A roofstructure as defined in claim 3 wherein each of said coordinates definesthe location on said ring of the connection between said ring and atleast one of said structural members.
 4. f/d b/a which pass through thefollowing coordinates with respect to the major (x) and minor (y) axesthereof: x + or - d y + or - (e(1+2+. . . (n-1)+n)+f)x + or -(d+nc)y +or - (e(2+3+. . . (n-1)+n)+f)x + or - (d+c(n+(n+(n-1))y + or - (e(+3 . .. (n-1)+n)+f). . . . . . x + or -(d+c(n+(n-1 . . . +2+1)y + or - f witha and b being given and equal respectively to one-half of thepreselected length of the major and minor axes of the curve and c, d andn being arbitrarily selected, whereby the stresses in said ring aresubstantially funicular.
 5. A roof structure as defined in claim 4wherein said structural members are cables comprising tension supportmembers.
 6. A roof structure as defined in claim 5 including a membranesecured to said cables to enclose said roof structure.
 7. A roofstructure as defined in claim 4 wherein said structural members arerigid structural arches.
 8. A roof structure as defined in claim 7including another set of arches extending substantially parallel to atleast one of the major and minor axes of said curve.